Consider the Strawberry

In general, I hate doing this — because it feels like a self-promotional trick — but in order for this post to make any kind of sense, you have to go back and read the last one.  In particular, you have to read Max’s comment.  I will put on my teacher face and wait for a few minutes.


For two reasons, I’m going to unpack the strawberry analogy a bit more: (1) I am in love with it, and (2) it highlights an important pedagogical point about the relationship between squares and rectangles.  For serious.

As Max pointed out, even very small children have no problem recognizing the rather trivial fact that all strawberries are fruits even though not all fruits are strawberries.  On the flip side, anyone who has ever taught geometry knows, with something like absolute certainty, that much older and more mathematically savvy students have great difficulty recognizing that all squares are rectangles even though not all rectangles are squares.  The situations are structurally identical (in each case we have some set, X, which is a proper subset of another set, Y), but the second one is much more problematic.  Why might that be?

The seemingly obvious answer is that recognizing a strawberry is nearly automatic, and probably evolutionarily encoded, while recognizing a square requires abstract reasoning about the congruence of mathematical objects called “line segments.”  But I’m not at all convinced that’s the problem.  They are both ultimately pattern-recognition tasks.  Without language getting in the way, you (and small children) can probably recognize strawberries and squares with comparable facility.

Which brings us to the language.  Even though the strawberry:fruit::square:rectangle situations are structurally identical, there is an important (and subtle) linguistic distinction in the latter case.  Consider the following story.

You find your favorite small child/guinea pig and present a challenge.  In your left hand you hold a strawberry, and in your right an apple.  You say to this child, “Which hand has the fruit in it?”  The child blinks at you for several moments, trying to study your face for clues about the answer to what has just got to be a trick question, before finally, tentatively, reaching out to point at one of your hands, more or less at random.  You reward the child with a piece of delicious fruit.

Consider the same story, except now you hold in your left hand a picture of a square, and in your right a picture of a generic rectangle.  You say to this child, “Which hand has the rectangle in it?”  The child immediately points to your right hand.  You reward the child with, I guess, a delicious piece of rectangle.

Why are these stories so different?  I submit that it’s not a mathematical issue.  The real problem stems from the fact that, linguistically, there is no unprivileged fruit: every class of fruit gets its own name.  But “square” is privileged relative to “rectangle.”  When presented with a generic rectangle, we have no word for saying that it is “a rectangle that is not a square.”  In fact, I made up the phrase “generic rectangle” precisely to try and convey that information.

So it turns out I lied a little bit before (how fitting) when I said the fruit/rectangle situations were structurally identical.  It’s true that in each case we have a set (square, strawberry) that is a subset of a larger set (rectangle, fruit), but it turns out the larger sets have different linguistic partitions.

Rectangles = \{Rectangles \cap Squares\} \cup \{Rectangles \cap Squares^C\}

Fruits = Apples \cup Apricots \cdots \cup Strawberries \cup \cdots \cup Watermelons

So when you ask the child which hand contains the rectangle, she chooses the generic rectangle immediately.  Why?  Because, had you meant the square, then you damn sure would’ve just said “square” in the first place, even though both hands hold perfectly correct answers to your challenge.  If our language were set up such that strawberries were the only specially named fruits (which seems like something Max would wholeheartedly support), the child in the first story would likewise choose your non-strawberry hand every time, without hesitation.

So what can we do with this?  It seems that strawberries have something to teach us about squares.  Actually, it seems that all the other fruits have something to teach us about rectangles.  It’s taken the entire history of humanity to organize fruits into useful equivalence classes, but luckily we find ourselves in a much, much simpler situation with rectangles; after all, there are only two classes we care about!  We already have a name for squares, so let’s call non-square rectangles “nares.”  Now our partition looks like this:

Rectangles = Squares \cup Nares

Which hand has the nare in it?  Easy.  Better yet, unambiguous.  Now, I’m not seriously lobbying for the introduction of nares into the mathematical lexicon (for one thing, nare is already a word for a weird thing), but it might be a fun way to introduce young children to the concept of a non-square rectangle.  After removing the greatest impediment to understanding the square/rectangle relationship (that “square” is the lone special case of this broader class of “rectangles,” which word is generally reserved for “rectangles-but-not-squares,” since, if someone means “square,” we already have a freaking word for it), that scaffolding can eventually be disassembled.

But the cognitive edifice the scaffolding initially supported will have cured a little by then.  In other words, why not make the distinction we actually care about explicit from the beginning, rather than end up in linguistic contortions to get around the fact that the distinction is solely implicit in standard usage?  Make up your own word, I don’t care.  Don’t want to be cute about it?  Fine.  Just abbreviate non-square rectangles as NSRs or something.  But make them easy to talk about — as easy as it is to talk about a tangerine or cumquat rather than a “fruit that might be a strawberry, but very often is not.”  Because, seriously, if that’s the way our fruit classification worked, there would be an awful lot of kids running around with the reasonable and tightly-held belief that strawberries are not fruit.

And that would be a shame.

Inconvenient Truths

As happens with amazing frequency, Christopher Danielson said something interesting today on Twitter.

And, as also happens with impressive regularity, Max Ray chimed in with something that led to an interesting conversation — which, in the end, culminated in my assertion that not everything that is mathematically true is pedagogically useful.  I would go further and say that a truth’s usefulness is a function of the cognitive level at which it becomes both comprehensible and important — but not before.

By way of an example, Cal Armstrong took a shot at me (c.f. the Storify link above) for my #TMC13 assertion that it is completely defensible to say that a triangle (plus its interior) is a cone.  Because he is Canadian, I think he will find the following sentiment particularly agreeable: we’re both right.  A triangle both is, and is not, a cone, depending on the context.  It might be helpful to think of it as Schrödinger’s Coneangle: an object that exists as the superposition of two states (cone and triangle),  collapsing into a particular state only when we make a measurement.  In this case, the “measurement” is actually made by our audience.

When I am speaking to an audience of relative mathematical maturity, I can (ahem…correctly) say that cone-ness is a very broadly defined property: given any topological space, X, we can build a cone over X by forming the quotient space

CX := X \times [0, 1] / \sim

with the equivalence relation ~ defined as follows:

(x,1) \sim (y,1) := x-y \in X \times \{0\}

If we take X to be the unit interval with the standard topology, we get a perfectly respectable Euclidean triangle (and its interior).  Intuitively, you can think of taking the Cartesian product of the interval with itself, which gives you a filled-in unit square, and then identifying one of the edges with a single point.  Boom, coneangle.  Which, like Sharknado, poses no logical problems.

Of course, it is a problem when you’re talking to a middle school geometry student.  In that situation, saying that a triangle is a cone is both supremely unhelpful and ultimately dishonest.  What we really mean is that, in the particular domain of 3-dimensional Euclidean geometry, when we have a circle (disk) in a plane and one non-coplanar point, we can make this thing called a cone by taking all the line segments between the point and the base.  But to that student, in that phase of mathematical life, the particular domain is the only domain, and so we rightly omit the details.  In an eighth-grade geometry class, there is absolutely no good reason to introduce anything else.

 photo cone_zpsf6e0fb1f.gif

Constructing a topological cone over the unit interval

We do this all the time as math teachers.  “Here, kid, is something that you can wrap your head around.  It will serve you quite well for a while.  Eventually we’re going to admit that we weren’t telling you the whole story — maybe we were even lying a little bit — but we’ll refine the picture when you’re ready.  Promise.”

Which brings me back to Danielson’s tweet.  From a mathematical point of view, there are all kinds of problems with saying that a rectangle has “two long sides and two short sides” (so many that I won’t even attempt to name them).  But how bad is this lie?  Better yet, how bad is the spirit of this lie?  I think it depends on the audience.  I’m not sure it’s so very wrong to draw a sharp (albeit technically imaginary) distinction for young children between squares and rectangles that are not squares.  It doesn’t seem all that different to me, on a fundamental level, from saying that cones are 3-dimensional solids.  Or that you can’t take the square root of a negative number.  Or that the sum of the interior angles of a quadrilateral is 360 degrees.  None of those statements is strictly true, but the truths are so very inconvenient for learners grappling with the concepts that we actually care about at the time.  It’s not currently important that they grasp the complete picture.  And it’s probably not feasible for them to do so, anyway.

Teaching mathematics is an iterative process, a feedback loop.  New information is encountered, reconciled with existing knowledge, and ultimately assimilated into a more complete understanding.  Today you will “know” that squares and rectangles are different.  Later, when you’re ready to think about angle measure and congruence, you will learn that they are sometimes the same.  Today you will “know” that times can only be 0 if either a or b is zero.  And tomorrow you will learn about the ring of integers modulo 6.

I will tell you the truth, children.  But maybe not today.